0 = V_t(S,t) + C₂(S,t)*V_SS(S,t) + C₁(S,t)*V_S(S,t) + C₀(S,t)*V(S,t),    0 < S < Smax,    0 < t < T,

C₂(S,t) = +0.5*sigma^2(t)*S^2,    C₁(S,t) = +r(t)*S,    C₀(S,t) = - r(t),

The European put option gives the option holder the optional right to sell the underlying stock at price E at time T (on the other hand, the option writer is obliged to buy the underlying stock from the holder). However, the rational seller hopes that the market price S drops so the stock can be sold at the higher price E at time T to make a profit of E-S, otherwise there is no reason to exercise the right to sell.

Thus, the final boundary condition is called the put option payoff function,

V(S,T) = max[E - S, 0],

The opposite case is the European call option is the optional right to buy with profit S-E. Another class of options is named an American Option, which can be exercised at any time before or at T, so is more complicated with a interior condition, making a free boundary problem, due to early exercise, and the free (interior) boundary has to be found as part of the solution.

The zero stock price boundary condition arises from a natural boundary condition for the PDE at S = 0 combined with the final condition above,

V(0,t) = E exp(-R(t))   for E > 0 and   R'(t) = -r(t)   with   R(T) = 0,

In order to avoid an infinite stock price condition for the finite numerics, an artificial cutoff is used as an approximation to large stock price boundary condition,

V(S_max,t) = 0,

S_max=q*E

1. Apply the Crank Nicolson Implicit Method to the problem, with NS evenly spaced stock (S) subintervals on [0, q E] and
   NT evenly spaced time (t) subintervals on [0, T].
   • Note this problem is a backward parabolic partial differential equation, not like the forward parabolic equation (diffusion equation) discussed in class. The integration is from the final time T to the initial time 0, so you need to use the backward time ordering, j = NT-1:-1:0 starting from the final condition V_i,NT = V(S_i,T) for t(j) = j*delta(t).

   • To solve the implicit Crank-Nicolson algebraic equations for the unknowns, you can NOT use the efficient Thomas Tridiagonal Algorithm, due to 2nd order upwinding the drift term C₁(S,t)*V_S(S,t) for stability, so you have to use than an LU decomposition or regular forward Gaussian elimination. Second order upwinding needs to be used to preserve CNI O(Delta{S})²

   • Since CNI uses the midpoint approximation in time, all explicit time dependence in coefficients must be evaluated at t_j+0.5.

   • You may use a combination of C/Fortran programming and MATLAB or either one to solve this problem.

1. Model Parameter Values:

   
   E = $100, in US Dollars,
   r(t) = r0+r1*t/T, with test values r0 = 0.0325, i.e., starting from 3.25% interest, and r1 = -0.25 for test values. In your code only input concrete value once since you need to write professional general code, so use the variables r0, r1 etc. thereafter.
   sigma(t) = sig0 + sig1*t+sig2*t^2, with test values sig0 = 0.012, i.e., starting from 1.2% volatility (compare to CP2 MLS values) sig1 = 0.0138, and sig2 = - 0.00125.
   T = 4 years,

   
   completing the financial engineering or mathematical problem formulation.

2. Numerical Parameter Values:

   
   Compare two (2) mesh sets:
   Mesh Set 1: NS = 200,   NT = 100,  
   Mesh Set 2: NS = 400,   NT = 200,  

   q = 5.0,
   S₀=0

   
   completing the computational financial engineering problem formulation.

1. Cover Page.

2. Executive Summary: Itemized list briefly describing problem, results (including a figure or table important to your report) and recommendations from you conclusions for the Busy Boss.

3. Introduction: Motivation for the report.

4. Problem and Methods: Well-documented with descriptions of the problem, variable, major steps.

5. Output Tables: Well-labeled, including at least,
   • Table 1: Comparison of the put option price V(S,0) for nodes closest to S = $90, $80, $70 and $60.

6. Output Figures: Well-labeled, including at least,
   • Figure 1: 3-dimensional plots in (S,t,V) for the whole solution, for both mesh sets.
   • Figure 2: 2-dimensional plots in (S,V(S,0)) for the initial values, for both mesh sets.
   • Figure 3: 2-dimensional plots of (t, r(t)) and (t, sigma(t)).

     It may be convenient to output your programming results to MATLAB ready m-files, or it may be more convenient to use MATLAB for everything.

7. Discussion of Results: Give a good discussion of your results with regard to accuracy, efficiency, and the value dependence on stock price S and t.

8. Acknowledgements and References: Be a good scholar and document your sources of information, including consultation with others.

9. Conclusions: What did you learn from this project?

1. UIC PCLabs MATLAB available software. Also check out your departmental computers such as in EECS and MSCS.

2. MATLAB Help Page (Hanson).

3. F. B. Hanson and J. J. Westman, Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation, SIAM Books, to appear 2004, Chapter 12 Applications in Financial Engineering { Chapter 12 Draft}

4. (NRC): William H. Press et al., Numerical Recipes in C, C⁺⁺ or F: The Art of Scientific Computing, Cambridge U. Press. { Click on Numerical Recipes Home Page}

5. Paul Wilmott, Sam Howison and Jeff Dewynne, Mathematics of Financial Derivatives: A Student Introduction, Cambridge U. Press, 1996.

6. Paul Wilmott, Derivatives: The Theory and Practice of Financial Engineering, Wiley, 1998.

7. Google Black-Scholes, etc.